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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b2">F</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b2">F</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>  &amp; (  for <font color="Olive" title="b3">f</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span> holds  <font color="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b3">f</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">F</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>  &amp; (  for <font color="Olive" title="b2">f</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span> holds  <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b2">f</font> ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">S</font> =  <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S9M</font> = <font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">XxxS9M</font> =  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">XxS</font> =  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">XxS9xM</font> = <font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> $1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">F</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
 for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span> holds  <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">f</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></span>
<a NAME="E3:45"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E4:45"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F</font> or <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_1"/>
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:45_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">and </span><br/><a NAME="E4:45_1"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">f</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E3:45_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="waybel26.html#T2" target="_self" title="WAYBEL26:th.2">Th2</a></span>;<br/>
<a NAME="E6:45_1"/>
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E3:45_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E4:45_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c9">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T31" target="_self" title="WAYBEL26:th.31">Th32</a></span>;<br/>
<a NAME="E8:45_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:45_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E10:45_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c2">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T28" title="YELLOW_1:th.28">YELLOW_1:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:45_1"/>
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c8">F</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> $1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">G</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:45"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
 for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span> holds  <font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">f</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></span>
<a NAME="E8:45"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E9:45"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">G</font> or <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_2"/>
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:45_2"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">G</font>
 <span class="kw">and </span><br/><a NAME="E4:45_2"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:45_2"/>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c2">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E3:45_2"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></span>;<br/>
<a NAME="E6:45_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T28" title="YELLOW_1:th.28">YELLOW_1:28</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">f9</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:45_2"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f9</font>
 <span class="kw">and </span><br/><a NAME="E9:45_2"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E10:45_2"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
<font color="Maroon" title="c13">f9</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_2"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/>
<a NAME="E11:45_2"/>
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c13">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E3:45_2"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E4:45_2"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E8:45_2"><span class="lab"><font color="Green" title="E44">A10</font></span></a></span>;<br/>
<a NAME="E12:45_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">g</font> is   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45_2"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="ref" href="waybel26.html#T33" target="_self" title="WAYBEL26:th.33">Th34</a></span>;<br/>
<a NAME="E13:45_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T2" target="_self" title="WAYBEL26:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:45_2"/>
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">take </span>
<font color="Maroon" title="c9">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>  &amp; (  for <font color="Olive" title="b1">f</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span> holds  <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b1">f</font> ) )</span><br/>

<span class="lab"><font color="Green" title="E42">A13</font></span>:  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel18.html#D3" title="WAYBEL18:def.3">WAYBEL18:def 3</a></span>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T30" target="_self" title="WAYBEL26:th.30">Th31</a></span>
<br/>.= 
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T11" title="CARD_3:th.11">CARD_3:11</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">G</font> = <font color="Maroon" title="c10">G</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/>
<a NAME="E12:45"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T32" target="_self" title="WAYBEL26:th.32">Th33</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span><br/><a NAME="E1:45_4"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>
 ;<br/><a NAME="E2:45_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A15</font></span>: 
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:45"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E10:45"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="funct_6.html#T57" title="FUNCT_6:th.57">FUNCT_6:57</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> = 
<font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E2:45_4"><span class="lab"><font color="Green" title="E44">A15</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A16</font></span>: 
<font color="Maroon" title="c11">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>;<br/>
<a NAME="E15:45"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span></span>)</span>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel25.html#T14" title="WAYBEL25:th.14">WAYBEL25:14</a></span>;<br/>
<a NAME="E16:45"/><span class="lab"><font color="Green" title="E46">A18</font></span>: 
<font color="Maroon" title="c9">F</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">b</font> or <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_5"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">f9</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c15">g9</font> = <font color="Maroon" title="c13">b</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T2" target="_self" title="WAYBEL26:th.2">Th2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c13">b</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T1" target="_self" title="WAYBEL26:th.1">Th1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c18">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c18">i</font></span><br/><a NAME="E1:45_5_1"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">Fai</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c20">Fbi</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c18">i</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T1" target="_self" title="WAYBEL26:th.1">Th1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> implies  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Fbi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:45_5_1_1"/>
<font color="Maroon" title="c21">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Fbi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">x</font> = <font color="Maroon" title="c21">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:45_5_1_1"/>
( <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c17">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>;<br/><a NAME="E4:45_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A21</font></span>: 
<font color="Maroon" title="c20">Fbi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font></span>)</span> <a href="waybel18.html#K5" title="WAYBEL18:func.5">.</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:45"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E10:45"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="funct_6.html#T56" title="FUNCT_6:th.56">FUNCT_6:56</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">fx</font> = <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font>, <font color="Maroon" title="c24">gx</font> = <font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:45"><span class="lab"><font color="Green" title="E45">A17</font></span></a></span>;<br/><a NAME="E6:45_5_1_1"/>
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c16">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>;<br/><a NAME="E7:45_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A22</font></span>: 
<font color="Maroon" title="c19">Fai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font></span>)</span> <a href="waybel18.html#K5" title="WAYBEL18:func.5">.</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:45"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E10:45"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="funct_6.html#T56" title="FUNCT_6:th.56">FUNCT_6:56</a></span>;<br/><a NAME="E8:45_5_1_1"/>
<font color="Maroon" title="c16">f</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_5"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="waybel26.html#T3" target="_self" title="WAYBEL26:th.3">Th3</a></span>;<br/><a NAME="E9:45_5_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a></span>;<br/><a NAME="E10:45_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">fx</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c24">gx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:45"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></span>;<br/><a NAME="E11:45_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A23</font></span>: 
<font color="Maroon" title="c23">fx</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c18">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c24">gx</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></span>;<br/><a NAME="E12:45_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E13:45_5_1_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Fbi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_5_1_1"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="txt" href="#E4:45_5_1_1"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="txt" href="#E11:45_5_1_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:45_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">Fai</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c20">Fbi</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:45_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c18">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_5_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="ref" href="waybel26.html#T3" target="_self" title="WAYBEL26:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:45_5"/>
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E47">A24</font></span>:  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c2">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></span>
<br/>.= 
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T28" title="YELLOW_1:th.28">YELLOW_1:28</a></span>
;<br/>
<a NAME="E18:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A25</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T32" target="_self" title="WAYBEL26:th.32">Th33</a>, <a class="ref" href="funct_5.html#T56" title="FUNCT_5:th.56">FUNCT_5:56</a></span>;<br/>
<a NAME="E19:45"/><span class="lab"><font color="Green" title="E49">A26</font></span>: 
<font color="Maroon" title="c11">G</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">b</font> or <font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_7"/><span class="lab"><font color="Green" title="E50">A27</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T1" target="_self" title="WAYBEL26:th.1">Th1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_7_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> implies  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:45_7_1"/>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c16">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">fx</font> = <font color="Maroon" title="c14">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>, <font color="Maroon" title="c19">gx</font> = <font color="Maroon" title="c15">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:45"><span class="lab"><font color="Green" title="E45">A17</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_7_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">fx</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c20">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">gx</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c20">j</font></span><br/><a NAME="E1:45_7_1_1"/><span class="lab"><font color="Green" title="E51">A28</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">aj</font> = <font color="Maroon" title="c12">a</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c20">j</font>, <font color="Maroon" title="c22">bj</font> = <font color="Maroon" title="c13">b</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c20">j</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T1" target="_self" title="WAYBEL26:th.1">Th1</a></span>;<br/><a NAME="E3:45_7_1_1"/>
<font color="Maroon" title="c12">a</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c20">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">b</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c20">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_7"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></span>;<br/><a NAME="E4:45_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">aj</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c22">bj</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_7_1_1"><span class="lab"><font color="Green" title="E51">A28</font></span></a>, <a class="ref" href="waybel26.html#T3" target="_self" title="WAYBEL26:th.3">Th3</a></span>;<br/><a NAME="E5:45_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A29</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">aj</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">bj</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a></span>;<br/><a NAME="E6:45_7_1_1"/>
( <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c13">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>;<br/><a NAME="E7:45_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A30</font></span>: 
<font color="Maroon" title="c19">gx</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">bj</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:45"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="ref" href="funct_6.html#T56" title="FUNCT_6:th.56">FUNCT_6:56</a></span>;<br/><a NAME="E8:45_7_1_1"/>
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c12">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>;<br/><a NAME="E9:45_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">fx</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">aj</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:45"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="ref" href="funct_6.html#T56" title="FUNCT_6:th.56">FUNCT_6:56</a></span>;<br/><span class="kw">hence </span><a NAME="E10:45_7_1_1"/>
<font color="Maroon" title="c18">fx</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c20">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">gx</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c20">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_7_1_1"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E5:45_7_1_1"><span class="lab"><font color="Green" title="E52">A29</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:45_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">fx</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">gx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></span>;<br/><span class="kw">hence </span><a NAME="E6:45_7_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:45"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:45_7"/><span class="kw">then </span>
<font color="Maroon" title="c14">f</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:45_7"/>
<font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T3" target="_self" title="WAYBEL26:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p4">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span><br/><a NAME="E1:45_8"/>
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span>) &amp;  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="ref" href="waybel26.html#T32" target="_self" title="WAYBEL26:th.32">Th33</a>, <a class="ref" href="funct_5.html#T56" title="FUNCT_5:th.56">FUNCT_5:56</a></span>;<br/><a NAME="E2:45_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A31</font></span>: 
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T57" title="FUNCT_6:th.57">FUNCT_6:57</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> = 
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E2:45_8"><span class="lab"><font color="Green" title="E50">A31</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p4">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E21:45"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:45"/>
<font color="Maroon" title="c9">F</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E19:45"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="yellow16.html#T15" title="YELLOW16:th.15">YELLOW16:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span> holds  <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b1">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c12">f</font> be   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c12">f</font></span><br/>

<a NAME="E23:45"/>
<font color="Maroon" title="c12">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T2" target="_self" title="WAYBEL26:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:45"/>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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